M ay 2 00 2 SPHERE PACKINGS IV

1.1. The steps. The Kepler conjecture asserts that no packing of spheres in three dimensions has density greater than π/ √ 18 ≈ 0.74048. This paper is one of a series of papers devoted to the Kepler conjecture. This series began with [I], which proposed a line of research to prove the conjecture, and broke the conjecture into smaller conjectural steps which imply the Kepler conjecture. The steps were intended to be equal in difficulty, although some have emerged as more difficult than others. This paper completes part of the fourth step. The main result is Theorem 4.4. As a continuation of [F] and [III], this paper assumes considerable familiarity with the constructions, terminology, and notation from these earlier papers. See [F] for the definitions of quasi-regular tetrahedra, upright and flat quarters and their diagonals, anchors, Rogers simplices, standard clusters, standard regions, the Q-system, V -cells, local V -cells, and decomposition stars. We will use a number of constants and functions from [I] and [F]: δoct = (π − 2ζ)/ √ 8, pt = −π/3 + 2ζ, ζ = 2 arctan( √ 2/5), t0 = 1.255, φ(h, t), φ0 = φ(t0, t0), Γ(S) is the compression, vor(S) is the analytic Voronoi function, vor(·, t) and vor0 = vor(·, t0) are the truncated Voronoi functions, σ(S) is the score, τ(S) measures what is squandered by a simplex, dih(S) is the dihedral angle along the first edge of a simplex, sol is the solid angle, R(a, b, c) is a Rogers simplex with parameters a ≤ b ≤ c, and S(y1, . . . , y6) is a simplex with edge lengths yi with the standard conventions from [I] on the ordering of edges. The definitions of σ(S) and τ(S) are particularly involved. The definition depends on the structure and context of S [F.3]. At the heart of this approach is a geometric structure, called the decomposition star, constructed around the center of each sphere in the packing. A function σ, called the score, is defined on the space of all decomposition stars. An upper bound of 8 pt ≈ 0.4429 on the score implies the Kepler conjecture [F, Proposition 3.14]. A second function τ , measuring what is squandered, is defined on the space of decomposition stars. If a decomposition star squanders more than (4πζ − 8) pt ≈ 14.8 pt ≈ 0.819, then it scores less than 8 pt.